Two circles with centers $O,O'$ intersect at $A,B$. The line $TT'$ is tangent to the first circle at $T$ and to the second at $T'$. The perpendiculars from $T,T'$ meet $OO'$ at $S,S'$. Ray $AS$ meets the first circle again at $R$, and ray $AS'$ meets the second again at $R'$. Prove that $R,B,R'$ are collinear.
Let $n$ be a positive integer and let $a,b$ be coprime integers greater than $1$. Suppose $M=a^{6^n}+b^{6^n}$ has at least two odd divisors greater than $1$. Determine the remainder when $M$ is divided by $6\cdot12^n$.
For real numbers $a,b$, define $x_0,x_1,x_2,\ldots$ by $x_0=a$, $x_{n+1}=x_n+b\sin x_n$. If $b=1$, prove that the sequence converges to a finite limit for every $a$. If $b\gt 2$, prove that it diverges for some $a$.
Find the largest value of
$$\frac1{\sqrt x}+\frac2{\sqrt y}+\frac3{\sqrt z}$$
for positive numbers $x,y,z$ satisfying
$$\begin{cases}\dfrac1{\sqrt2}\le z\le\min\{x\sqrt2,y\sqrt3\},\\[3pt]x+z\sqrt3\ge\sqrt6,\\[3pt]y\sqrt3+z\sqrt{10}\ge2\sqrt5.\end{cases}$$
Let $a_1,a_2,\ldots,a_{2n}$ be a permutation of $\{1,2,\ldots,2n\}$ such that
$$|a_i-a_{i+1}|\ne|a_j-a_{j+1}|$$
for all distinct indices for which these adjacent differences are defined. Prove that $a_1=a_{2n}+n$ if and only if $1\le a_{2i}\le n$ for every $i=1,2,\ldots,n$.